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[Octave-cvsupdate] SF.net SVN: octave:[9968] trunk/octave-forge/main/queueing
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From: <mma...@us...> - 2012-03-19 12:28:05
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Revision: 9968
http://octave.svn.sourceforge.net/octave/?rev=9968&view=rev
Author: mmarzolla
Date: 2012-03-19 12:27:55 +0000 (Mon, 19 Mar 2012)
Log Message:
-----------
documentation improvements
Modified Paths:
--------------
trunk/octave-forge/main/queueing/doc/markovchains.txi
trunk/octave-forge/main/queueing/inst/ctmc_bd.m
trunk/octave-forge/main/queueing/inst/dtmc_bd.m
Modified: trunk/octave-forge/main/queueing/doc/markovchains.txi
===================================================================
--- trunk/octave-forge/main/queueing/doc/markovchains.txi 2012-03-19 09:34:32 UTC (rev 9967)
+++ trunk/octave-forge/main/queueing/doc/markovchains.txi 2012-03-19 12:27:55 UTC (rev 9968)
@@ -39,7 +39,8 @@
@iftex
@tex
-$$P\left(X_{n+1} = x_{n+1}\ |\ X_n = x_n, X_{n-1} = x_{n-1}, \ldots, X_0 = x_0 \right) = P\left(X_{n+1} = x_{n+1}\ |\ X_n = x_n\right)$$
+$$\eqalign{P\left(X_{n+1} = x_{n+1}\ |\ X_n = x_n, X_{n-1} = x_{n-1}, \ldots, X_0 = x_0 \right) \cr
+& = P\left(X_{n+1} = x_{n+1}\ |\ X_n = x_n\right)}$$
@end tex
@end iftex
@ifnottex
@@ -52,10 +53,10 @@
The evolution of a Markov chain with finite state space @math{@{1, 2,
@dots{}, N@}} can be fully described by a stochastic matrix @math{{\bf
-P}(n) = P_{i,j}(n)} such that @math{P_{i, j}(n) = P( X_{n+1} = j\ |\
+P}(n) = [ P_{i,j}(n) ]} such that @math{P_{i, j}(n) = P( X_{n+1} = j\ |\
X_n = i )}. If the Markov chain is homogeneous (that is, the
transition probability matrix @math{{\bf P}(n)} is time-independent),
-we can simply write @math{{\bf P} = P_{i, j}}, where @math{P_{i, j} =
+we can simply write @math{{\bf P} = [P_{i, j}]}, where @math{P_{i, j} =
P( X_{n+1} = j\ |\ X_n = i )} for all @math{n=0, 1, @dots{}}.
The transition probability matrix @math{\bf P} must satisfy the
@@ -244,7 +245,8 @@
@iftex
@tex
-$$P(X_{t_{n+1}} = x_{n+1}\ |\ X(t_n) = x_n, X(t_{n-1}) = x_{n-1}, \ldots, X(t_0) = x_0) = P(X(t_{n+1}) = x_{n+1}\ |\ X(t_n) = x_n)$$
+$$\eqalign{P(X(t_{n+1}) = x_{n+1}\ |\ X(t_n) = x_n, X(t_{n-1}) = x_{n-1}, \ldots, X(t_0) = x_0) \cr
+&= P(X(t_{n+1}) = x_{n+1}\ |\ X(t_n) = x_n)}$$
@end tex
@end iftex
@ifnottex
@@ -255,8 +257,9 @@
@emph{infinitesimal generator matrix} @math{{\bf Q} = [Q_{i,j}]} such
that for each @math{i \neq j}, @math{Q_{i, j}} is the transition rate
from state @math{i} to state @math{j}. The elements @math{Q_{i, i}}
-must be defined in such a way that the infinitesimal generator matrix
-@math{\bf Q} satisfies the property @math{\sum_{j=1}^N Q_{i, j} = 0}.
+qre defined as @math{Q_{i, i} = - \sum_{j \neq i} Q_{i, j}}, such that
+matrix @math{\bf Q} satisfies the property that, for all @math{i},
+@math{\sum_{j=1}^N Q_{i, j} = 0}.
@DOCSTRING(ctmc_check_Q)
@@ -373,8 +376,8 @@
@end ifnottex
@noindent where @math{{\bf \pi}(t) = {\bf \pi}(0) \exp({\bf Q}t)} is
-the state occupancy probability at time @math{t}; @math{\exp({\bf A})}
-is the matrix exponential of @math{\bf A}.
+the state occupancy probability at time @math{t}; @math{\exp({\bf Q}t)}
+is the matrix exponential of @math{{\bf Q}t}.
@DOCSTRING(ctmc_exps)
Modified: trunk/octave-forge/main/queueing/inst/ctmc_bd.m
===================================================================
--- trunk/octave-forge/main/queueing/inst/ctmc_bd.m 2012-03-19 09:34:32 UTC (rev 9967)
+++ trunk/octave-forge/main/queueing/inst/ctmc_bd.m 2012-03-19 12:27:55 UTC (rev 9968)
@@ -17,37 +17,51 @@
## -*- texinfo -*-
##
-## @deftypefn {Function File} {@var{Q} =} ctmc_bd (@var{birth}, @var{death})
+## @deftypefn {Function File} {@var{Q} =} ctmc_bd (@var{b}, @var{d})
##
## @cindex Markov chain, continuous time
## @cindex Birth-death process
##
-## Returns the @math{N \times N} infinitesimal generator matrix @math{Q}
-## for a birth-death process with given rates.
+## Returns the infinitesimal generator matrix @math{Q} for a continuous
+## birth-death process over state space @math{1, 2, @dots{}, N}.
+## @code{@var{b}(i)} is the transition rate from state @math{i} to
+## @math{i+1}, and @code{@var{d}(i)} is the transition rate from state
+## @math{i+1} to state @math{i}, @math{i=1, 2, @dots{}, N-1}.
##
-## @strong{INPUTS}
+## Matrix @math{\bf Q} is therefore defined as:
##
-## @table @var
+## @iftex
+## @tex
+## $$ \pmatrix{ -\lambda_1 & \lambda_1 & & & & \cr
+## \mu_1 & -(\mu_1 + \lambda_2) & \lambda_2 & & \cr
+## & \mu_2 & -(\mu_2 + \lambda_3) & \lambda_3 & & \cr
+## \cr
+## & & \ddots & \ddots & \ddots & & \cr
+## \cr
+## & & & \mu_{N-2} & -(\mu_{N-2}+\lambda_{N-1}) & \lambda_{N-1} \cr
+## & & & & \mu_{N-1} & -\mu_{N-1} }
+## $$
+## @end tex
+## @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and
+## death rates, respectively.
+## @end iftex
+## @ifnottex
+## @example
+## @group
+## / \
+## | -b(1) b(1) |
+## | d(1) -(d(1)+b(2)) b(2) |
+## | d(2) -(d(2)+b(3)) b(3) |
+## | |
+## | ... ... ... |
+## | |
+## | d(N-2) -(d(N-2)+b(N-1)) b(N-1) |
+## | d(N-1) -d(N-1) |
+## \ /
+## @end group
+## @end example
+## @end ifnottex
##
-## @item birth
-## Vector with @math{N-1} elements, where @code{@var{birth}(i)} is the
-## transition rate from state @math{i} to state @math{i+1}.
-##
-## @item death
-## Vector with @math{N-1} elements, where @code{@var{death}(i)} is the
-## transition rate from state @math{i+1} to state @math{i}.
-##
-## @end table
-##
-## @strong{OUTPUTS}
-##
-## @table @var
-##
-## @item Q
-## Infinitesimal generator matrix for the birth-death process.
-##
-## @end table
-##
## @end deftypefn
## Author: Moreno Marzolla <marzolla(at)cs.unibo.it>
Modified: trunk/octave-forge/main/queueing/inst/dtmc_bd.m
===================================================================
--- trunk/octave-forge/main/queueing/inst/dtmc_bd.m 2012-03-19 09:34:32 UTC (rev 9967)
+++ trunk/octave-forge/main/queueing/inst/dtmc_bd.m 2012-03-19 12:27:55 UTC (rev 9968)
@@ -17,68 +17,83 @@
## -*- texinfo -*-
##
-## @deftypefn {Function File} {@var{P} =} dtmc_bd (@var{birth}, @var{death})
+## @deftypefn {Function File} {@var{P} =} dtmc_bd (@var{b}, @var{d})
##
## @cindex Markov chain, discrete time
## @cindex Birth-death process
##
-## Returns the @math{N \times N} transition probability matrix @math{P}
-## for a birth-death process with given rates.
+## Returns the transition probability matrix @math{P} for a discrete
+## birth-death process over state space @math{1, 2, @dots{}, N}.
+## @code{@var{b}(i)} is the transition probability from state
+## @math{i} to @math{i+1}, and @code{@var{d}(i)} is the transition
+## probability from state @math{i+1} to state @math{i}, @math{i=1, 2,
+## @dots{}, N-1}.
##
-## @strong{INPUTS}
+## Matrix @math{\bf P} is therefore defined as:
##
-## @table @var
+## @iftex
+## @tex
+## $$ \pmatrix{ (1-\lambda_1) & \lambda_1 & & & & \cr
+## \mu_1 & (1 - \mu_1 - \lambda_2) & \lambda_2 & & \cr
+## & \mu_2 & (1 - \mu_2 - \lambda_3) & \lambda_3 & & \cr
+## \cr
+## & & \ddots & \ddots & \ddots & & \cr
+## \cr
+## & & & \mu_{N-2} & (1 - \mu_{N-2}-\lambda_{N-1}) & \lambda_{N-1} \cr
+## & & & & \mu_{N-1} & (1-\mu_{N-1}) }
+## $$
+## @end tex
+## @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and
+## death probabilities, respectively.
+## @end iftex
+## @ifnottex
+## @example
+## @group
+## / \
+## | 1-b(1) b(1) |
+## | d(1) (1-d(1)-b(2)) b(2) |
+## | d(2) (1-d(2)-b(3)) b(3) |
+## | |
+## | ... ... ... |
+## | |
+## | d(N-2) (1-d(N-2)-b(N-1)) b(N-1) |
+## | d(N-1) 1-d(N-1) |
+## \ /
+## @end group
+## @end example
+## @end ifnottex
##
-## @item birth
-## Vector with @math{N-1} elements, where @code{@var{birth}(i)} is the
-## transition probability from state @math{i} to state @math{i+1}.
-##
-## @item death
-## Vector with @math{N-1} elements, where @code{@var{death}(i)} is the
-## transition probability from state @math{i+1} to state @math{i}.
-##
-## @end table
-##
-## @strong{OUTPUTS}
-##
-## @table @var
-##
-## @item P
-## Transition probability matrix for the birth-death process.
-##
-## @end table
-##
## @end deftypefn
## Author: Moreno Marzolla <marzolla(at)cs.unibo.it>
## Web: http://www.moreno.marzolla.name/
-function P = dtmc_bd( birth, death )
+function P = dtmc_bd( b, d )
if ( nargin != 2 )
print_usage();
endif
- ( isvector( birth ) && isvector( death ) ) || \
+ ( isvector( b ) && isvector( d ) ) || \
usage( "birth and death must be vectors" );
- birth = birth(:); # make birth a column vector
- death = death(:); # make death a column vector
- size_equal( birth, death ) || \
+ b = b(:); # make b a column vector
+ d = d(:); # make d a column vector
+ size_equal( b, d ) || \
usage( "birth and death vectors must have the same length" );
- all( birth >= 0 ) || \
+ all( b >= 0 ) || \
usage( "birth probabilities must be >= 0" );
- all( death >= 0 ) || \
+ all( d >= 0 ) || \
usage( "death probabilities must be >= 0" );
- all( ([birth; 0] + [0; death]) <= 1 ) || \
- usage( "Inconsistent birth/death probabilities");
- ## builds the infinitesimal generator matrix
- P = diag( birth, 1 ) + diag( death, -1 );
+ all( ([b; 0] + [0; d]) <= 1 ) || \
+ usage( "d(i)+b(i+1) must be <= 1");
+
+ P = diag( b, 1 ) + diag( d, -1 );
P += diag( 1-sum(P,2) );
endfunction
%!test
%! birth = [.5 .5 .3];
%! death = [.6 .2 .3];
-%! fail("dtmc_bd(birth,death)","Inconsistent");
+%! fail("dtmc_bd(birth,death)","must be");
%!demo
%! birth = [ .2 .3 .4 ];
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